Ms Higgins reached a compensation agreement with the Commonwealth in December 2022 after she launched legal action against her employers in the previous coalition government.
Ms Higgins alleged she was raped by Bruce Lehrmann in 2019 inside the office of then-coalition minister Linda Reynolds, who they both worked for.
Mr Lehrmann has always denied Ms Higgins' allegation.
This week text messages between Ms Higgins and her partner David Sharaz emerged suggesting the pair contemplated strategising her story with Senator Gallagher, when she was in opposition, after the rape allegation was made public.
It's further alleged the senator misled parliament in 2021 when she rebuffed claims she was tipped off about the rape allegation.
Mr Albanese on Friday defended Senator Gallagher, saying the claim against her was bizarre.
"You had allegations by a Liberal staffer that another Liberal staffer had a sexual assault in a Liberal minister's office, and somehow, somehow Katy Gallagher has some responsibility for what was going on here," he told Seven's Sunrise program.
"Like this is bizarre.
"The idea that there's this conspiracy somehow ... that's nonsense."
Deputy Prime Minister Richard Marles also defended Senator Gallagher saying she had "made her position very clear".
However, opposition finance spokeswoman Jane Hume said clarifying the context of the text messages was important.
"There are some questions to be answered," she told Today.
"We shouldn't run a running commentary on the cases that are going on, because there are processes still under way.
"The Sofronoff inquiry hasn't reported yet."
The Sofronoff inquiry into the handling of Mr Lehrmann's subsequent rape trial is due to be handed down in July.
Meanwhile, Senator Reynolds - who was criticised over her handling of Ms Higgins' situation - has threatened to refer her compensation payment to the National Anti-Corruption Commission, which comes into force on July 1, with the support of her leader Peter Dutton.
But Independent MP Zali Steggall on Friday accused the coalition of turning Ms Higgins into a political punching bag to further their attacks on the Labor government.
"This is a pretty vile attack from Dutton and Linda Reynolds, putting at risk the mental health of someone that's very vulnerable," Ms Steggall told Today.
"We have to remember that Brittany Higgins is a very vulnerable young woman and making her the punching bag of a political attack is really unsavoury."
Ms Steggall called for the content of emails and phone calls made by Senator Reynolds after she was made aware of the alleged assault to be made public and also defended Senator Gallagher.
"In my experience, Katy Gallagher has been a very ethical person in all my dealings with her," she said.
"This is quite a distraction from the Liberal Party, and it says a lot for Dutton's moral compass."
Mr Dutton has doubled down on the issue of compensation, believed to be up to $3 million and paid to Ms Higgins after she lodged a successful personal injury claim.
"Linda Reynolds is a person of great honour and she feels rightly aggrieved in the process here," he said on Thursday.
"I suspect this would be one of the first issues that the integrity commission would deal with."
Prosecutors ultimately dropped the charge against Mr Lehrmann because of fears about the impact of a second rape trial on Ms Higgins' mental health after his initial trial was abandoned because of juror misconduct.
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